Sea Quark and Gluon Polarization in the Nucleon

538 Brazilian Journal of Physics, vol. 37, no. 2B, June, 2007 

Sea Quark and Gluon Polarization in the Nucleon 

D. de Florian, G. A. Navarro, and R. Sassot 

Departamento de Física, Universidad de Buenos Aires, 

Ciudad Universitaria, Pab.1 (1428) Buenos Aires, Argentina 

Received on 29 August, 2006 

We present results on the quark and gluon polarization in the nucleon obtained in a combined next to leading 

order analysis to the available inclusive and semi-inclusive polarized deep inelastic scattering data. Using the 

Lagrange multiplier method, we asses the uncertainty inherent to the extraction of the different spin dependent 

parton densities in a QCD global fit, and the impact of the increased set of semi-inclusive data now available. 

Keywords: Semi-Inclusive DIS; Perturbative QCD 

I. INTRODUCTION 

For more than fifteen years, polarized inclusive deep inelastic 

scattering (pDIS) has been the main source of information 

on how the individual partons in the nucleon are polarized at 

very short distances. 

Many alternative experiments have been conceived to improve 

this situation. The most mature among them are those 

based on polarized semi-inclusive deep inelastic scattering 

(pSIDIS), i.e. a pDIS experiment where a particular hadron 

is tagged in the hadronic final state. Choosing different target 

and final state hadrons, the respective cross sections are sensitive 

to different combinations of flavored quarks and antiquarks, 

to be disentangled. These experiments began with the 

pioneering efforts of SMC, in the late nineties [1], followed 

by those of HERMES at DESY [2], and lately COMPASS at 

CERN [3], and are planned to be improved at the Thomas Jefferson 

National Laboratory (JLAB) in the near future [4]. 

The phenomenological impact of the pSIDIS data proved 

to be encouraging although scarce in the initial stage: the reduced 

number of data and the relatively large estimated errors, 

at best allowed to check the consistency between pDIS and 

pSIDIS in a variety of spin-flavor symmetry scenarios [5–7]. 

With the availability of larger sets of pSIDIS data, much more 

precise, and for final state hadrons and targets of different flavor 

composition, the situation now has changed dramatically. 

pSIDIS data have a non negligible weight in combined global 

fits at present, comparable to that of inclusive data, and also 

show clear preferences for the light sea quark polarization. It 

also helps to constrain the strange sea quark and gluon polarization 

complementing the information already obtained from 

pDIS. 

In the following we present results obtained in a combined 

next to leading order analysis to the recently updated set of 

pDIS and pSIDIS data [8]. Specifically, we focused on the extraction 

of sea quark and gluon densities, analyzing the constraining 

power of the data on the individual densities. As result, 

we found not only a complete agreement between pDIS 

and pSIDIS data, but a very useful complementarity, leading 

to rather well constrained densities. 

Using the Lagrange multiplier approach [9], we explored 

the profile of the χ 2 function against different degrees of polarization 

in each parton flavor. In this way we obtained estimates 

for the uncertainty in the net polarization of each fla- 

vor, and in the parameters of the pPDFs. We compared results 

obtained with the two most recent sets of fragmentation 

functions. The differences are found to be within conservative 

estimates for the uncertainties. Nevertheless, there is a clear 

preference for a given set of FF over the other, shown in a difference 

of several units in the χ 2 of the respective global fits. 

In NLO global fits the overall agreement between theory and 

the full set of data is sensibly higher than in LO case. 

Finally, we analyze the behavior of the cross section for 

longitudinally polarized proton-proton collisions into neutral 

pions with a wide range of pPDFs sets coming from a rather 

conservative uncertainty interval. This observable is found to 

be crucially sensitive to the polarized gluon density and therefore 

an invaluable tool. We compute the required precision to 

be reached in the programed experiments in order to constrain 

even further this distribution and also future sets of pPDFs. 

A similar analysis is made for forthcoming pSIDIS data to be 

obtained at JLAB. 

II. GLOBAL FIT 

In our analysis [8], we followed the same conventions and 

definitions for the polarized inclusive asymmetries and parton 

densities adopted in references [6, 7], however we used more 

recent inputs, such as unpolarized parton densities [10] and 

the respective values for αs. In the totally inclusive case, the 

spin dependent asymmetries are given by: 

A N 1 (x,Q2 ) = gN 1 (x,Q2 ) 

F N 1 (x,Q2 ) = 

gN 1 (x,Q2 ) 

FN 2 (x,Q2 )/2x[1 + RN (x,Q2 , (1) 

)] 

where the inclusive spin-dependent nucleon structure function 

g N 1 (x,Q2 ) can be written at NLO as a convolution between polarized 

parton densities for quarks and gluons, ∆qi(x,Q 2 ) and 

∆g(x,Q 2 ), respectively, and coefficient functions ∆Ci(x)[11]; 

F N 1 (x,Q2 ) is the unpolarized nucleon structure function that 

can be written in terms of F N 2 (x,Q2 ) and R, the ratio of the 

longitudinal to transverse cross section [12]. 

Analogously, for the semi-inclusive asymmetries we have: 

A Nh 

1 (x,Q2 ) |Z � 

� 

Z dzgNh 1 (x,z,Q2 ) 

� 

Z dzFNh 1 (x,z,Q2 , (2) 

)

D. de Florian, G. A. Navarro, and R. Sassot 539 

TABLE I: Inclusive and semi-inclusive data used in the fit. 

Collaboration Target Final state # points Refs. 

EMC proton inclusive 10 [16] 

SMC proton, deuteron inclusive 12, 12 [17] 

E-143 proton, deuteron inclusive 82, 82 [18] 

E-155 proton, deuteron inclusive 24, 24 [19] 

Hermes proton,deuteron,helium inclusive 9, 9, 9 [2] 

E-142 helium inclusive 8 [20] 

E-154 helium inclusive 17 [21] 

Hall A helium inclusive 3 [22] 

COMPASS deuteron inclusive 12 [3] 

SMC proton,deuteron h + , h − 24, 24 [1] 

Hermes proton, deuteron, helium h + , h − , π + , π − , K + , K − , K T 36,63,18 [2] 

Total 478 

TABLE II: χ 2 values and first moments for distributions at Q 2 = 10 GeV 2 

set χ 2 χ 2 DIS χ2 SIDIS δu δd δs δg δΣ 

KRE 430.91 206.01 224.90 -0.0487 -0.0545 -0.0508 0.680 0.284 

NLO KKP 436.17 205.66 230.51 0.0866 -0.107 -0.0454 0.574 0.311 

KRE 457.54 213.48 244.06 -0.0136 -0.0432 -0.0415 0.121 0.252 

LO KKP 448.71 219.72 228.99 0.0497 -0.0608 -0.0365 0.187 0.271 

where the superscript h denotes the hadron detected in the final 

state, and the variable z is given by the ratio between the 

hadron energy and that of the spectators in the target. The 

region Z, over which z is integrated, is determined by kinematical 

cuts applied when measuring the asymmetries. For 

the spin dependent structure function gN 1 (x,Q2 ), we use the 

NLO expression [13] 

Fragmentation functions were taken from either [14] or 

[15], respectively. We also used the flavor symmetry and flavor 

separation criteria proposed in [14], at the respective ini- 

tial scales Q 2 i 

. The data sets analyzed include only points with 

Q 2 > 1 GeV 2 , listed in Table I, and totaling 478 data points. 

In Table II we summarize the results of the best NLO and 

LO global fits to all the data listed in Table I. We present fits 

obtained using alternatively fragmentation functions from reference 

[14], labeled as KRE, and from reference [15], labeled 

as KKP. 

Since the fit involves 20 parameters, the number of degrees 

of freedom for these fits is 478-20=458. Consequently, the 

χ 2 values obtained are excellent for NLO fits and very good 

for LO. The better agreement between theory and experiment 

found at NLO, highlights the importance of the corresponding 

QCD corrections, for the present level of accuracy achieved by 

the data. 

In NLO fits there seems to be better agreement when using 

KRE fragmentation functions. The difference between the total 

χ 2 values between KRE and KKP NLO fits comes mainly 

from the contributions related to pSIDIS data, while those associated 

to inclusive data are almost the same, as one should 

expect in a fully consistent scenario. 

Table II includes also the first moment of each flavor distribution 

at Q 2 = 10 GeV 2 , and that for the singlet distribution 

δΣ, as reference. Most noticeably, while the KRE NLO fit 

favors the idea of a SU(3) symmetric sea, KKP NLO finds u 

polarized opposite to d and to s. Gluon and strange sea quark 

polarization are similar in both fits and the total polarization 

carried by quarks is found to be around 30%. 

In Figures 1 and 2 we show the inclusive and semi-inclusive 

asymmetries computed with the different parameterizations 

both at LO and NLO accuracy, against the corresponding data 

sets. The differences between the various sets can hardly be 

noticed in the comparison to inclusive data in Figure 1, although 

are more significant when comparing to pSIDIS data, 

specially in the case of proton targets. This is due to the 

fact that the main difference between the sets are the light 

sea quark densities, which are probed by pSIDIS processes 

of proton targets. The pSIDIS asymmetries for deuterium targets 

are, of course, less sensitive to these differences since 

they average u and d contributions. 

III. UNCERTAINTIES 

Many strategies have been implemented in order to assess 

the uncertainties in PDFs and their propagation to observables, 

specially those associated with experimental errors in 

the data. The Lagrange multiplier method [9] probes the uncertainty 

in any observable or quantity of interest relating the 

range of variation of one or more physical observables dependent 

upon PDFs to the variation in the χ 2 used to judge the 

goodness of the fit to data. In Figure 3 we show the outcome 

of varying the χ 2 of the NLO fits to data against the first moment 

of the respective polarized parton densities δq at Q 2 = 10 

GeV 2 , one at a time. This is, to minimize 

Φ(λq,aj) = χ 2 (a j) + λq δq(a j) q = u,u,d,d,s,g. (3)


1 

0.5 

0 

1 

0.5 

0 

0.5 

0 

0.5 

0 

0.5 

0 

0.5 

0 

0.5 

0 

10 -2 

10 -2 

1 10 -2 

1 10 -2 

FIG. 1: Inclusive asymmetries computed with LO and NLO pPDFs against the corresponding data. 

10 -2 

1 10 -2 

1 10 -2 

1 10 -2 

FIG. 2: Semi-inclusive asymmetries computed with LO and NLO pPDFs against the corresponding data. 

In order to see the effect of the variation in χ 2 on the parton 

distributions themselves, in Figure 4, we show KRE best fit 

densities together with the uncertainty bands corresponding to 

∆χ 2 = 1 (darker band) and ∆χ 2 = 2% (light shaded band). As 

expected, the relative uncertainties in the total quark densities 

and those strange quarks are rather small. For gluon densities 

the ∆χ 2 = 1 band is also small, but the most conservative 

∆χ 2 = 2% estimate is much more significative. For light sea 

1 

quarks the ∆χ 2 = 1 bands are moderate but the ∆χ 2 = 2% are 

much more larger. The dot line corresponds to each respective 

unpolarized parton densities for that values of Q 2 and x, 

showing that the positivity constraint is fulfilled.


χ 2 

470 

460 

450 

440 

430 

470 

460 

χ 450 2 

440 

430 

KRE NLO 

KKP NLO 

χ 2 

χINC (a) 

-0.5 0 -0.5 0 -0.1 0 

δu 

δd 

δs 

(d) 

0.5 1 -0.9 -0.5 -0.1 

δu+δu 

δd+δd 

(b) 

(e) 

0 1 

FIG. 3: χ 2 profiles for NLO fits obtained using Lagrange multipliers at Q 2 = 10 GeV 2 . 

IV. FUTURE PROSPECTS 

One of the measurements most eagerly awaited by the 

spin physics community is that of single inclusive large pT 

pion production in longitudinally polarized proton-proton collisions, 

which is right now being run at BNL RHIC [23]. The 

spin dependent asymmetry associated to this kind of process, 

Aπ0 LL is defined, as usual, in terms of the ratio between the polarized 

and the unpolarized cross sections, 

A π0 

LL = d∆σp p→π0 X 

dσ p p→π0 X 

which is strongly dependent on the gluon polarization. 

The data obtained up to now by the PHENIX Collaboration 

suggest a very small asymmetry, consistent with pPDFs sets 

with a moderate gluon polarization. In the following we apply 

the Lagrange multiplier method in order to explore the range 

of variation of the estimates for this asymmetry associated to 

the uncertainty in the present extraction of pPDFs. 

In Figure 5 we show the range of variation of Aπ0 LL at an 

intermediate value for pT = 6 GeV obtained with different sets 

of pPDFs against the variation of the χ2 to pDIS and pSIDIS 

data for these distributions. The profile of χ2 defines a well 

defined range of values for Aπ0 LL allowed by present pPDFs. 

The solid line represents the profile of χ2 obtained using KRE 

FFs, both in the global fit to data and in the computation of the 

asymmetry. The dashed line represents the same but for KKP 

FFs. The minima corresponding to both profiles are very close 

(4) 

δg 

(c) 

(f) 

χ 2 χ0+5% χ 2 χ0+2% χ 2 χ0+1 χ 2 χ0+5% χ 2 χ0+2% χ 2 χ0+1 suggesting a cancellation of the associated uncertainty. Notice 

that both the extraction of the pPDFs, and also the estimate 

of Aπ0 LL with a given set, rely on a set of FFs. The double 

dependence of the observable on the set of FFs used may had, 

in principle, even potentiated the disagreement. 

In Figure 5 we have included also the profiles obtained 

using δg instead of Aπ0 LL in minimization as a dotted line in 

the case of KRE and a dashed dotted line for KKP. Clearly, 

the sea quark polarization can conspire in order to yield a 

larger/smaller asymmetry than the one obtained with maximum/minimum 

gluon fit at a given χ2 , effect which in much 

more apparent at larger values of Aπ0 LL . This feature will have 

to be taken into account in the future for a very precise measurement 

of the gluon polarization. 

Unfortunately, the data collected so far in the first two runs 

of the PHENIX detector at RHIC have estimated errors much 

larger than the uncertainties in the values Aπ0 LL coming from 

the fits, however this situation is going to change dramatically 

in the near future. In Figure 6 we plot Aπ0 LL as a function 

of pT using the best NLO fits coming from KRE and 

KKP FFs (solid lines and dashes, respectively) and also with 

KRE variants designed to enhance/reduce Aπ0 LL with ∆χ2 = 2% 

(dotted-dashed lines). We have also included the expected uncertainties 

for the next two runs centered at KRE best fit estimate 

[24]. Clearly, the future measurement of the asymmetry 

would certainly be able to constrain even further the gluon and 

therefore reduce the uncertainties in pPDFs. 

Another source of complementary information to further 

constrain the extraction of pPDFs and also FFs is the exper-


χ 2 

455 

450 

445 

440 

435 

430 

0.4 

0.2 

0 

-0.2 

0.06 

0.04 

0.02 

0 

-0.02 

-0.04 

-0.06 

x(Δu+Δu – 

) 

x(Δd+Δd – ) 

xΔu – 

10 -2 

x Bj 

xΔu v 

xΔd v 

xΔd – 

10 -2 

x Bj 

xΔg – 

xΔs – 

KRE (NLO) 

KKP (NLO) 

unpolarized 

KRE χ 2 

KRE χmin +1 

KRE χ 2 

KRE χmin +2% 

FIG. 4: Parton densities at Q 2 = 10 GeV 2 , and the uncertainty bands corresponding to ∆χ 2 = 1 and ∆χ 2 = 2% 

p T =6 GeV 

KRE NLO 

KKP NLO 

-0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01 

A π 0 

ALL FIG. 5: Profile of the global χ 2 to data against A π0 

LL at pT = 6 GeV 

imental program of the E04-113 experiment at JLAB, which 

propose to measure pion and kaon pSIDIS asymmetries for 

proton and deuteron targets [4]. In Figure 7a we show the 

profile of χ 2 of the global fits using KRE (solid line) and 

A π 

0.03 

0.02 

0.01 

0 

ALL 0 

-0.01 

Run 5 

2006-7 

10 -2 

KRE NLO 

KRE NLO Δχ 2 =2% 

KKP NLO 

x Bj 

0.4 

0.2 

0 

-0.2 

0.06 

0.04 

0.02 

0 

-0.02 

-0.04 

-0.06 

-0.02 

0 2 4 6 8 10 12 14 

p (GeV/c) 

T 

FIG. 6: A π0 

LL vs. pT computed with different pPDF sets and the estimated 

errors expected by PHENIX at RHIC. 

KKP (dashes) FFs against the π − pSIDIS asymmetry on proton 

targets at one of the kinematic configurations of E04-113 

(xB j = 0.203, Q 2 = 2.3 GeV 2 < z >= 0.5). The expected


455 

450 

445 

χ 

440 

2 

435 

430 

χ 2 +5% 

χ 2 +2% 

χ 2 +1 

0.2 0.3 0.4 

A π- 

Ap (a) 

KRE NLO 

KKP NLO 

A K- 

0 0.2 0.4 

Ap FIG. 7: Profile of the global χ 2 to data against A π− 

p and A K− 

p proposed to be measured by E04-113 at JLAB. 

uncertainty for the measurement of this asymmetry shown in 

the bottom of the figure, is significantly smaller than those of 

the previous measurements, and also smaller than the present 

uncertainty coming from the fit. In Figure 7b we show the 

same as in the previous figure but but for negative kaons. 

This asymmetry has not been measured yet and the difference 

between the predictions coming form KRE (solid line) and 

KKP (dashes) sets is much larger that the expected uncertainties. 

The measurement of this last observable together with 

the combined effect of data for different targets (proton and 

deuteron) and final state hadrons (positive and negative pions 

and kaons) will certainly constrain the fit even further, specifically 

the sea quark densities, and at the same time provide a 

more stringent test on the quark flavor separation for the FFs 

used in the analysis. 

V. CONCLUSIONS 

The availability of an important set of new data on polarized 

processes together with the appropriate theoretical tools 

required to interpret them, obtained and developed in the last 

few years, have led the extraction of pPDFs in the proton 

to mature and to become an important source of information 

which will still keep growing in the near future. 

In this paper, we have assessed the feasibility of obtaining 

pPDFs, with special emphasis on the sea quark densities, from 

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(b) 

455 

450 

445 

440 

435 

430 

a combined NLO QCD analysis of pDIS and pSIDIS data. We 

have estimated the uncertainties associated to the extraction 

of each parton density, finding a well constrained scenario. 

pSIDIS data is not only consistent with pDIS, but improves 

the constraining power of the fit for all the distributions, being 

crucial for the light sea quarks. 

The overall picture found for the quark densities at a 10 

GeV 2 is one in which, within uncertainties, up quarks are almost 

100% polarized parallel to the proton, down quarks antiparallel 

in a similar proportion, and sea quarks have a small 

flavor symmetric negative polarization [8, 25] 

Two programmed experiments, the one based on the 

PHENIX detector already running at RHIC, and the E04-113 

experiment at JLab will be able to reduce dramatically the uncertainty 

in both the gluon and the light sea quark densities 

respectively, the latter providing also an even more stringent 

test on fragmentation functions. 

Acknowledgments 

Partially supported by CONICET, Fundación Antorchas, 

UBACYT and ANPCyT, Argentina. We warmly acknowledges 

M. B. Gay Ducati, and the Local Comitee of the I Latin 

American workshop on High Energy Phenomenology for their 

hospitality during the meeting where this talk was presented. 

[3] A. Bressan, for the COMPASS Collaboration, hep-ex/0501040. 

[4] X. Jiang et al. Jefferson Lab experiment E04-113: Semi- 

Inclusive Spin Asymmetries on the Nucleon Experiment (hepex/0412010).


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Sea Quark and Gluon Polarization in the Nucleon

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